No AV receiver can have two different HDMI inputs selected at the same time. A few will let you watch video from on an HDMI input while listening to audio from an analogue (or maybe SPDIF) source at the same time, but probably not your Yamaha RX-V371.

Are you using a TV with multiple HDMI inputs? You can use one for the PS2, another for the AVR.

You also might consider getting an HDMI matrix switch. Monoprice has them for under $60 with 4 inputs & 2 outputs. They're available elsewhere, too.

You could use the matrix switch for all (or most) of your source devices or just for the PS2, switching the source devices between the TV and the AVR.

E.G. you could connect the PS2 to the matrix switch could be switched between the TV and the AVR. Switch the PS2 to the TV and select an appropriate (audio) input on the AVR, or switch the PS2 to the AVR and watch and listen simultaneously.

Or you could just swap cables manually, although that can cause excessive wear on the connectors.

p.s.
Bear in mind that most AVRs introduce lag in both audio and video -- more processing = more lag. A direct connection to the display is probably the best for gaming, if at all possible. A switch should not introduce any lag, since it does no signal processing.